Civil Engineering Reference
In-Depth Information
To incorporate geometric nonlinearity into the analysis with material nonlinearity based on
the FAM, the stiffness matrices
K
i
,
K
0
i
, and
K
0
i
of the
i
th column member must be derived by
taking into consideration the applied axial compressive force on the member. Though the
theory of using stability functions in analyzing elastic frame buckling was fully developed half
a century ago, the derivation of the elastic stiffness matrix
K
i
is repeated here mainly because
the equations of the deflected shapes are important to the derivation of inelastic
K
0
i
and
K
0
i
matrices. In addition, the derivation of the deflected shapes also highlights the difference
between the stability functions approach and the current approximate method of geometric
nonlinear analysis using the geometric stiffness approach and the
P
-
Δ
approach.
By recognizing that only column members will be subjected to the axial compression
while beam members will experience insignificant axial force due to the presence of the slab,
distributed loading along the members due to gravity will therefore be ignored throughout the
derivation of the stiffness matrices. In addition, the columns are rotated 90 degrees in the der-
ivation using the classical beam theory with an applied axial compressive load, and therefore
the word 'beam' will be used interchangeably with the word 'column' in this section.
7.2.1 Stiffness Matrix
[
K
i
]
Consider separately the four cases of a beam deflection subjected to various displacement
patterns as shown in Figure 7.2 in deriving the stiffness matrices. Here,
V
1
i
,
m
1
i
,
V
2
i
, and
m
2
i
represent the fixed-end shears and moments of the beam, and
i
=1,…, 4 represents the four
cases of unit displacement patterns of beam deflection.
Using the classical Bernoulli-Euler beam theory with “plane sections remain plane”, where
the moment is proportional to the curvature, the governing equilibrium equation describing the
deflection of the beam member can be written as
EIv
0
ð Þ
00
+
Pv
00
=0
ð7
:
14Þ
where
E
is the Young's modulus,
I
is the moment of inertia,
v
is the lateral deflection, and
P
is
the axial compressive force of the member, and prime represents taking derivatives of the
corresponding variable with respect to the
x
-direction of the member. By assuming
EI
is
constant along the member, the solution to the fourth-order ordinary differential equation is:
v
=
A
sin
kx
+
B
cos
kx
+
Cx
+
D
ð7
:
15Þ
y
y
P
θ
=1
m
21
m
22
EI
EI
ν=1
m
11
P
x
P
P
x
Case 1
Case 2
m
12
V
21
V
12
V
22
V
11
m
23
y
P
y
m
24
EI
Case 4
ν
=1
x
P
x
P
P
Case 3
m
13
V
23
m
14
EI
V
13
V
14
V
24
θ
=1
Figure 7.2
Displacement patterns and the corresponding fixed-end forces.
